Computer Algebra Systems (CAS)

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Computer Algebra Systems (CAS) are software programs designed to perform symbolic mathematics, enabling manipulation of mathematical expressions in a way similar to human reasoning. They can simplify, solve, and analyze mathematical problems symbolically rather than numerically, facilitating tasks in algebra, calculus, and other areas of mathematics.

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Key research themes

1. How can Computer Algebra Systems (CAS) and Digital Mathematical Libraries (DML) be jointly leveraged for mutual verification and reliability enhancement?

This research theme explores approaches to verify and improve the accuracy, consistency, and reliability of both CAS and digital mathematical libraries by translating and cross-validating mathematical expressions between them. Such verification is crucial because both CAS and DMLs are widely used in mathematical research and applications, yet both are prone to errors. The interplay leveraging their overlap can reveal discrepancies which may be due to translation errors, CAS implementation bugs, or inaccuracies in the libraries, thus fostering mutual improvement.

Comparative Verification of the Digital Library of Mathematical Functions and Computer Algebra Systems

by Bela Gipp

2022, Tools and Algorithms for the Construction and Analysis of Systems

Key finding: Developed an improved translation tool (LACAST) capable of converting mathematical expressions from the NIST Digital Library of Mathematical Functions (DLMF) to CAS syntax (Maple and Mathematica), doubling the range of... Read more

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2. What are mathematics students’ evolving beliefs about using Computer Algebra Systems, and how stable are these beliefs over time?

This theme investigates senior secondary students' attitudes and beliefs about CAS as they learn to use them in routine and advanced mathematical problem-solving. Understanding the stability of these beliefs is important because attitudes towards CAS influence students’ engagement, effective use, and learning outcomes. The theme addresses how students perceive CAS usefulness, correctness, ease of use, and its relationship to traditional pen-and-paper approaches.

The stability of mathematics students’ beliefs about working with CAS

by Vicki Steinle

2023, Mathematics Education Research Journal

Key finding: Through repeated measures tracking twelve Year 11 students, found that students’ beliefs regarding CAS (including usefulness, correctness of answers, and appropriateness compared to pen-and-paper methods) were generally... Read more

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3. How are Computer Algebra Systems integrated and utilized within secondary and higher mathematics education curricula, and what are the implications for teaching and assessment?

This theme examines pedagogical strategies, curriculum design, and assessment frameworks involving CAS in mathematics education at secondary and undergraduate levels. It addresses practical challenges such as technological availability, teacher preparedness, and student engagement, alongside the redesign of curriculum content to exploit CAS capabilities. It also studies the impact of CAS-enabled examinations and transitions to technology-active assessment formats.

Examinations in the Final Year of Transition to Mathematical Methods Computer Algebra System (CAS)

by David Leigh-Lancaster

2025, merga.net.au

Key finding: Analyzed large-scale student performance data from parallel CAS-enabled and traditional cohorts in Victorian Year 12 mathematics examinations, revealing that CAS cohorts often matched or slightly outperformed non-CAS cohorts... Read more

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Mathematical Methods Computer Algebra System (CAS) 2004 pilot examinations and links to a broader research agenda

by David Leigh-Lancaster

2025, … : Research, Theory and …

Key finding: Reported on longitudinal pilot CAS implementation showing sustained student performance parity or superiority on common examination items compared to standard courses, and highlighted increasing adoption of CAS... Read more

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Computer Algebra Systems in the Mathematics Curriculum -Report of the invitation MEI seminar supported by Texas Instruments CONTENTS Background

by David Leigh-Lancaster

2021

Key finding: Presented a case study using Maple to equip undergraduate computer science students with algebraic tools for complex analysis tasks beyond their usual training, such as average-case algorithmic analysis. Demonstrated that... Read more

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Teaching Mathematics Using Mathematica

by Paul C Abbott

2016

Key finding: Showcased the use of Mathematica as a CAS to teach advanced mathematical concepts through symbolic manipulation, functional programming, and visualizations, notably implementing Gram-Schmidt orthogonalization. Highlighted how... Read more

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Computer Algebra Systems in the Mathematics Curriculum -Report of the invitation MEI seminar supported by Texas Instruments CONTENTS Background

by David Leigh-Lancaster

2021

Key finding: Emphasized the vital role of CAS tools in computer science curricula, particularly for algorithmic reasoning and average-case analyses, which traditionally are mathematically demanding. The integration of CAS software (e.g.,... Read more

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All papers in Computer Algebra Systems (CAS)

Moment-Based Density Approximants

by Serge B. Provost

2023

It is often the case that the exact moments of a statistic of the continuous type can be explicitly determined, while its density function either does not lend itself to numerical evaluation or proves to be mathematically intractable. The... more

Figure 1. Exact and approximate (dashed line) PDFs. [Pq or Pq] in the Appendix]

Figure 2. Exact and approximate (dashed line) CDFs. [PQ in the Appendix]

As Figure 3 indicates, the exact and approximate CDFs differ by less than 0.001 over the interval [0, 3] in this case. Figure 3. The difference between the exact and approximate CDFs. [Qd_ in the Appendix]

Figure 4. Exact and approximate (dashed line) PDFs. [Pb in the Appendix] Consider a mixture of two equally weighted beta distributions with parameters (3, 2) and (2, 30), respectively. A fifteenth-degree polynomial approximation was obtained from the compact formula given in equation (14). The exact density function of this mixture and its approximant, both plotted in Figure 4, are manifestly in close agreement. A glance at the Mathematica code that is provided in the Appendix for this example should convince the reader that very little programming is indeed required. Clearly, methodologies that are based on only a few moments would fail to provide satisfactory approximations in this case.

where Gamma[.] =IT[.] denotes the gamma function or, observing that LaguerreL|j, vy, X] with X* replaced by yx [k] as defined earlier, is equivalent to LaguerreL[j, v, (Y — a) /c] with Y* replaced by py[&], we obtain equation (24) and making the change of variable Y =cX +a, we obtain the following density approximant for Y:

Figure 5. Exact and approximate (dashed line) CDFs. [CDFLN in the Appendix]

Figure 6. Exact density function and initial gamma approximant. [PGE in the Appendix] Figure 6 shows the exact density function of the mixture as well as the initial gamma density approximation given by fy, [y]. Clearly, traditional approximants which make use of three or four moments could not capture adequately all the distinctive features of this particular distribution.

Figure 7. Exact and approximate (dashed line) PDFs. [PDEA in the Appendix] The exact density function, fy[y], and its approximant, fy,,[y], evaluated from equation (29), are plotted in Figure 7. As pointed out in Remark 3.1, this density approximant results from a polynomial adjustment applied to the shifted gamma density specified by fy, [y]. (Once such an approximant is obtained, a spline could be fitted in order to reduce the degree of precision that would be required in subsequent calculations.)

Figure 8. Exact and approximate (dashed line) PDFs. [S5 in the Appendix] Example 1 is revisited by making use of a beta density as a base density function in Result 4.1 or equivalently by resorting to an approximant expressed in terms of Jacobi polynomials. Since the base density function already provides a good approximation in this case, fewer moments are required than in the case of the Legendre polynomial approximant (eight as opposed to 13) in order to achieve a similar degree of accuracy.

Figure 9. PDF of U for n= 2 (dashed line) and n = 20. [P6 in the Appendix] Figures 9 and 10 show the approximate density and distribution functions of L for N = 12, gq; =2, q2 =4, and p=2 when one lets =2 (dashed line) anc n = 20 (solid line). Mathai [25] determined that for N - 4) -— q2 = 6, q, = 2, anc p =2, the 95th and 99th percentiles of the distribution are respectively 0.87825 and 0.94719. As evaluated in the Appendix, Fly,, [0.87825] = 0.950001 anc F1y,, [0.94719] = 0.990001, where F1y, [w] denotes a percentile corresponding tc the point w, which is approximated on the basis of m moments. It is seen from th¢ graph that an adequate approximant can be obtained from two moments in thi case. In fact, Fly, [0.87825] = 0.9507 and F Ly, [0.94719] = 0.9903.

Figure 10. CDF of U for n= 2 (dashed line) and n= 20. [Cé in the Appendix] Approximants Based on Hermite Polynomials We can approximate densities whose tail behaviour is congruent to that of a normal density function by means of the modified Hermite polynomials given by

Figure 11. Exact and approximate (dashed line) PDFs. [S7, S71, or S7a in the Appendix] Consider an equally weighted mixture of two Gaussian distributions with parame- ters (3,4) and (1, 1), whose density function is plotted as a solid line in Figure 11. On the basis of 30 moments, identical density approximants (represented by a dashed line) are obtained whether we make use of equation (42) or Result 4.1.

from equations (13) and (15) are given by fy,[y] and fly, [y], respectively. The exact and approximate cumulative distribution functions obtained by integratior are denoted by Q,-[u] and Qy. [w], respectively.

We use the notation introduced in Section 4. First, we approximate the density function of the random variable V as defined in Section 2 by making use of equation (42) with T;[x] = G[a + B+1,a@+1, x], a modified Jacobi polynomial. To conform to the general notation used in Section 4, we will again replace V with Y.

Identical density approximants denoted by fly, [y] and f2y,[y] in the following code are obtained from Result 4.1 and equation (43), respectively.

‘The same density approximant is obtained from equation (32). We use the notation introduced in Section 4. First, we approximate the density function of the Gaussian mixture with fy, [y] by making use of an alternate representation of equation (42) with 7;[x] = H*;[x], a modified Hermite polyno- mial. The function fly, [y] provides an approximant which is directly expressed in terms of the moments of Y.

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Towards precise, scalable and automatic analysis of analog and mixed signal circuits

by Osman Hasan

2022, 2010 International Conference on Information and Emerging Technologies

These days Analog and Mixed signal (AMS) circuits are widely being designed and used in many safety and financial critical application domains. This fact brings the challenge of efficiently and accurately analyzing these AMS circuits as... more

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Supplementary Material: Ecological Public Goods Games:Cooperation and Bifurcations: Jacobian at Q

by Christoph Hauert

2022, Theor. Pop. Biol.

Definitions JacobianMatrix@f_List ? VectorQ, x_ListD := Outer@D, f, xD ê; Equal üü HDimensions êü 8f, x<L Equations of Motion (b=0) G@q_, z_D := 8-q z H1-qL F@zD,-H1-zL Hz q Hr-1L H1z n-1 L-dL< F@z_D := 1 + Hr-1L z n-1-r H1z n L... more

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Propuesta de una estrategia para la enseñanza de tópicos de computación a estudiantes de licenciatura en matemática

by Jorge León Díaz

2022

Campo de investigación: educación continua. Nivel educativo: superior Palabras clave: programación, TIC, matemática, enseñanza El presente trabajo tiene como propósito hacer una propuesta de estructuración de contenidos a enseñar sobre... more

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Towards precise, scalable and automatic analysis of analog and mixed signal circuits

by Nadra Ramzan

2022, 2010 International Conference on Information and Emerging Technologies

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Analytical solutions of elastic regions weakened by elliptic cracks

by Alain Carmasol

2022

The present study deals with the analytic resolution of fundamental problems of the plane elasticity theory. Based on the general framework suggested by Kolossov and Muskhelishvili through the complex representation theory, the... more

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On the Perimeter of an Ellipse

by Paul C Abbott

2021, The Mathematica Journal

physics.uwa.edu.au/~paul Introducing the parameter j into the Cartesian coordinates, as Hx = a sinHjL, y = b cosHjLL, one verifies that the ellipse equation is satisfied.

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On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion

by Peter Supancic and

2021, The Mathematica Journal

This article presents a compact analytic approximation to the solution of a nonlinear partial differential equation of the diffusion type by using Bürmannʼs theorem. Expanding an analytic function in powers of its derivative is shown to... more

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On Bürmann’s Theorem and Its Application to Problems of Linear and Nonlinear Heat Transfer and Diffusion

by Harald Schöpf

2021, The Mathematica Journal

Here are the coefficients B, as they are calculated with this procedure in their explicit ana- lytic form, according to the definition given in (2). Series [Biirmann[#*5 &, Sinh, {z, 0, 25}], {z, 0, 25}]

A useful application of the recursion (5)—(7) to the case of the expansion of the inverse function y(z) of f(z) around z = z follows immediately: we have y(f(z)) = z and it fol- lows that the inverse function is the Biirmann series of z in powers of f(z) — f(Z) (see 181). By writing A useful application of the recursion (5)—(7) to the case of the expansion of the inverse

Using the results listed in (10) for the Biirmann coefficients, we arrive at once at the expan- sion around Zp: w[z_] := Birmann[Log[1+H#] &, 1/ (1+#) &, {z, 0, 11}]

m The Generalized Form of Burmann Series Based on a Combinatorial Approach

While the expansion is valid for complex-valued functions, the plot shows only the real part of f. The Explicit Expression for the Coefficients 8,,(/; ¢, zo) in Terms of the Coefficients R;,(¢*, m)

The function fbiir shows how to apply (23) and (24) to the expansion of f(z) = 2° in The formula for the Biirmann coefficient 6,(f; 4,7) in terms of the coefficients R,(¢*, m) is thus given by

As an example, we calculate an expansion of arcsin[z] up to order 15 according to (14) ir Use this definition for numerical purposes, such as for plotting. = The Generalized Form of Burmann Series

A function calculating the expansion (32) is given below. To show that this approach is superior to a common Taylor expansion in a plot, we calculate the power series in z up to order 10. The result of this calculation performed up to order nine in © (@, 0, z) is

The plot shows that the series in (32) has only a small constant offset error for larger values of z, whereas the Taylor expansion dramatically deviates for smaller values of z, although it converges uniformly for all values of z. The series in (32) converges uniformly for all z and gives the exact value for the error function. The rearrangement of terms, leading to erfg(z), is thus justified. Even for the lowest order, we will find a result that shows no unbounded error, unlike the Taylor series. The order v+1 of first nonvanishing derivative of 6(z) at Zo is 2

m Application of Burmann Series to Problems of Nonlinear Heat Transfer This section applies the concept of Biirmann series to get solutions of nonlinear differen- tial equations. After a short introduction, an example from the field of the diffusion type is presented.

While the first boundary condition in (35) is regular (i.e. T (0, t) = Ts and 6(0) = a), the second one is given in terms of the asymptotic expression lim,,,. T(z, t) = T,. The equiv- alent value o of the derivative (d@/d1),-o has to be estimated, which is performed by calculating the time evolution of the total thermal energy of the semi-infinite half space [18]. The approximate value of this energy integral (in terms of an algebraic expression) is determined in appendix B, where we use the Biirmann series (10) to approximate the energy integral. The explicit expression is displayed in appendix B, equation (56), which describes the dependence of o on the parameters with a relative accuracy of 0.37%. alent value o of the derivative (d@/d1),-9 has to be estimated, which is performed by

If we consider y in (40) and (41) as a free parameter, we have to investigate how its choice influences the convergence of the corresponding Biirmann series. We write, using (37) and omitting the index y for w, All integrands are representable by uniformly convergent series expansions in powers of w <1, and thus the argument is similar to the one used in proving the convergence of the system (39). Furthermore, we observe that the expansion of the exponential function exp(-2 (B/ yy f dO...) converges more quickly for higher values of y. Thus, as long as y can be chosen to guarantee the condition Convergence of the Enhanced Burmann Series

We display the explicit expression for the third-order approximation below: If there is a solution y,; to equation (43) simultaneously fulfilling condition (44), we get an

This defines the cubic approximation according to (45)-(48).

The plot shows the numerically exact temperature profiles (the colored curves) and the exact solution’s third-order approximation according to equation (48) (the dotted lines), in the range z = 0-0.25 mat t = 10, 30, 100, 300, 1000, 3000, 10000 s. The plot shows the numerically exact temperature profiles (the colored curves) and the

Rearranging equation | of (50) in increasing powers ¢*”), we have two conditions The coefficients R;(f, m), defined by Expansion Coefficients of f(z)~” m Appendix A

The Recursive Formula for Burmann Coefficients C,(/; 4, Zo) This section shows how to calculate the coefficients of the Biirmann series recursively, which is easier to handle than (3) and more efficient when translated to symbolic pro- grams. If we use (1) and (2) to find convergent series representations of solutions to differ- ential equations, it is important to simplify the algorithms necessary to determine the expansion coefficients. This section shows how to calculate the coefficients of the Biirmann series recursively,

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LA DOCENCIA DE LOS CIRCUITOS ELÉCTRICOS EN LOS ALBORES DEL SIGLO XXI

by José Antonio Aquino

2019

Resumen-Se vive en la actualidad una vertiginosa carrera en la innovación tecnológica, y como protagonista de estos hechos se tienen a los ingenieros, los cuales requieren ser preparados en las universidades con las más avanzadas... more

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PROPUESTA PARA EL USO SOFTWARE DE ÁLGEBRA SIMBÓLICA EN LOS PROYECTOS DE FIN DE CARRERA EN INGENIERÍA MECATRÓNICA

by José Antonio Aquino

2018

In this paper, the use of computational symbolic algebra is proposed for the development of end-of-career projects in mechatronic engineering. Looking for that engineering students who are in the last section of their studies, are enabled... more

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Algorithm to compute the rank and a Cartan subalgebra of a matrix Lie algebra with Mathematica

by Candido martin gonzalez

2016

We present in this paper a set of routines constructed to compute the rank of a matrix Lie algebra and also to determine a Cartan subalgebra from a given list of elements

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Algorithm to compute the rank and a Cartan subalgebra of a matrix Lie algebra with Mathematica

by Candido martin gonzalez

2016

We present in this paper a set of routines constructed to compute the rank of a matrix Lie algebra and also to determine a Cartan subalgebra from a given list of elements.

Finally, we present the Lie algebra (see the references) so(4,4)@so(2,2). This Lie algebra is a model we have created that explains all the quarks and their corresponding antiquarks. We define a basis by making

and the a generic element is The rank is 4, and a Cartan subalgebra: Another example could be

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Teaching Mathematics Using Mathematica

by Paul C Abbott

2016

The Mathematica computer algebra system (CAS) is an attractive medium for teaching mathematics. New features in Mathematica 3.0 include editable typeset mathematical expressions; customizable palette interface; automatic... more

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Ingeniero en Biónica como líder del siglo XXI

by Sarai Vizuet

2016

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Responsabilidad social del ingeniero mecatrónico

by Guillermo Iván Esparza Padilla

2015

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Matrix Algebra Package for Mathematica

by Tomas McKelvey and

2015